Problem: $ -1.\overline{23} \div 3.\overline{21} = {?} $
Explanation: First convert the repeating decimals to fractions. $\begin{align*} 100x &= -123.2324...\\ x &= -1.2324...\end{align*} $ $\begin{align*} 99x &= -122 \\ x &= -\dfrac{122}{99}\end{align*} $ $\begin{align*} 100y &= 321.2121...\\ y &= 3.2121...\end{align*} $ $\begin{align*} 99y &= 318 \\ y &= \dfrac{318}{99}\end{align*} $ So, the problem becomes: $ -\dfrac{122}{99} \div \dfrac{318}{99} = {?} $ Dividing by a fraction is the same as multiply by the reciprocal of that fraction. $ -\dfrac{122}{99} \times \dfrac{99}{318} = {?} $ $ \phantom{-\dfrac{122}{99} \times \dfrac{318}{99}} = \dfrac{-122 \times 99}{99 \times 318} $ $ \phantom{-\dfrac{122}{99} \times \dfrac{318}{99}} = \dfrac{-122 \times \cancel{99}} {\cancel{99} \times 318} $ $ \phantom{-\dfrac{122}{99} \times \dfrac{318}{99}} = -\dfrac{122}{318} $ Simplify: ${= -\dfrac{61}{159}}$